Latitude of a Traverse Leg

A traverse is a chain of measured lines, and before any of it can be plotted each line is resolved into a north-south component (the latitude) and an east-west one (the departure). With azimuths measured clockwise from north, the north component is L cos α, which puts the cosine on the northing — the opposite of the schoolroom habit of putting cosine on x. The sign takes care of itself: azimuths between 90° and 270° return a negative latitude, meaning the course runs south.

The Great Trigonometrical Survey of India, begun in 1802, is the monument to this arithmetic. Every one of the Great Arc's stations was reduced through latitudes and departures computed by hand, by human "computers" working in ledgers, with theodolites weighing half a tonne carried up purpose-built towers; the survey ran for most of a century and its accumulated closure over 2,400 km was measured in metres. A worked example: a 250 ft course at azimuth 60° has a latitude of 250 cos 60° = 125 ft north. The trap is feeding a bearing where an azimuth belongs — N 60° E and azimuth 60° agree, but S 60° E is azimuth 120°.